Subgroups — #12968942

Order 1280.

1770 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10617626	C₂	1	2	yes	maximal
2	C₂×C₅×#405	C₂	1	1	yes	maximal, characteristic
3	#10615483	C₂	1	2	yes	maximal
4	C₂×#65395	C₂	1	2	yes	maximal
5	#10610277	C₂	1	8	yes	maximal, complemented
6	#10610276	C₂	1	8	yes	maximal, complemented
7	#10610210	C₂	1	4	yes	maximal, complemented
8	#10610173	C₂	1	4	yes	maximal, complemented
9	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
10	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₅×#117	C₂²	1	1	yes	characteristic
12	C₂×#4072	C₂²	1	4	yes	—
13	#65853	C₂²	1	16	yes	—
14	#65851	—	2	8	no	—
15	#65834	—	2	4	no	—
16	#65817	—	2	4	no	—
17	#65804	C₂²	1	8	yes	—
18	C₅×#405	C₂²	1	4	yes	—
19	C₅×#405	C₂²	1	4	yes	—
20	C₂×C₅×#97	C₂²	1	2	yes	—
21	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
22	#65410	C₂²	1	16	yes	—
23	#65395	C₂²	1	4	yes	—
24	#65395	C₂²	1	8	yes	—
25	#65395	C₂²	1	4	yes	—
26	#65395	C₂²	1	4	yes	—
27	#65395	C₂²	1	4	yes	—
28	C₂×#3950	C₂²	1	4	yes	—
29	C₂×#3949	C₂²	1	4	yes	—
30	#64991	C₂²	1	32	yes	complemented
31	#64972	C₂²	1	8	yes	complemented
32	#64972	C₂²	1	8	yes	complemented
33	#64971	C₂²	1	8	yes	complemented
34	#64969	C₂²	1	8	yes	complemented
35	#60660	—	5	1	no	maximal, Sylow
36	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
37	C₂×C₅×M₁₆	Q₈	1	8	yes	complemented
38	C₂×C₅×M₁₆	C₂³	1	4	yes	—
39	C₂²×C₅×C₈	Q₈	1	8	yes	complemented
40	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
41	C₂²×C₅×C₈	C₂³	1	2	yes	—
42	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
43	C₂²×C₅×C₈	C₂³	1	2	yes	—
44	C₂²×C₅×C₈	C₂³	1	2	yes	—
45	C₅×#117	D₄	1	4	yes	—
46	C₅×#117	C₂³	1	4	yes	—
47	C₂×C₅×C₄⋊C₄	D₄	1	2	yes	—
48	C₂×C₅×C₂²⋊C₄	D₄	1	2	yes	—
49	C₂×C₄²×C₅	C₂³	1	2	yes	—
50	C₂×C₂²⋊Dic₅	—	2	4	no	—
51	#4072	—	2	4	no	—
52	#4072	—	2	4	no	—
53	#4072	C₂³	1	8	yes	—
54	#4072	C₂³	1	8	yes	—
55	C₂×C₄⋊Dic₅	C₂³	1	8	yes	—
56	#4070	—	2	16	no	—
57	C₂×C₄×Dic₅	—	2	4	no	—
58	#4036	—	2	16	no	—
59	#4034	—	2	16	no	—
60	#4024	—	2	32	no	—
61	#4016	C₂³	1	16	yes	—
62	#4015	C₂³	1	16	yes	—
63	C₄×Dic₁₀	—	2	16	no	—
64	C₅×#97	C₂³	1	8	yes	—
65	C₄×C₅×C₈	C₂³	1	4	yes	—
66	C₄×C₅×C₈	C₂³	1	4	yes	—
67	#3950	C₂³	1	8	yes	—
68	#3950	C₂³	1	8	yes	—
69	#3950	C₂³	1	16	yes	—
70	#3949	C₂³	1	8	yes	—
71	#3949	C₂³	1	8	yes	—
72	#3949	C₂³	1	16	yes	—
73	#3458	—	5	2	no	—
74	#3164	—	5	8	no	—
75	#3163	—	5	8	no	—
76	#3162	—	5	4	no	—
77	#3160	—	5	4	no	—
78	#2911	—	5	2	no	—
79	C₂×#427	—	5	2	no	—
80	C₂×#405	D₅	1	1	yes	characteristic
81	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₅	C₂×D₄	1	4	yes	—
83	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₅	C₂×Q₈	1	4	yes	—
86	C₂²×C₄×C₅	C₂×Q₈	1	4	yes	—
87	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
88	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
89	C₂²×Dic₅	—	2	8	no	—
90	C₂×Dic₁₀	—	2	16	no	—
91	C₅×M₁₆	C₂×Q₈	1	16	yes	—
92	C₂×C₅×C₈	C₂×Q₈	1	4	yes	—
93	C₂×C₅×C₈	C₂×Q₈	1	4	yes	—
94	C₂×C₅×C₈	C₂×Q₈	1	8	yes	—
95	C₂×C₅×C₈	C₂×Q₈	1	8	yes	—
96	C₂×C₅×C₈	C₂×Q₈	1	8	yes	—
97	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
98	C₂×C₅×C₈	C₂⁴	1	2	yes	—
99	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
100	C₂×C₅×C₈	C₂⁴	1	4	yes	—
101	C₅×C₄⋊C₄	C₂×D₄	1	8	yes	—
102	C₅×C₂²⋊C₄	C₂×D₄	1	8	yes	—
103	C₄²×C₅	C₂×D₄	1	4	yes	—
104	C₄²×C₅	C₂⁴	1	4	yes	—
105	C₂²⋊Dic₅	—	2	8	no	—
106	C₂²⋊Dic₅	—	2	8	no	—
107	C₄⋊Dic₅	—	2	16	no	—
108	C₄⋊Dic₅	—	2	8	no	—
109	C₄⋊Dic₅	C₂⁴	1	16	yes	—
110	#683	—	2	32	no	—
111	C₄×Dic₅	—	2	8	no	—
112	C₄×Dic₅	—	2	8	no	—
113	C₂²×M₁₆	C₂×D₅	1	1	yes	characteristic
114	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
115	#544	—	5	16	no	—
116	#544	—	10	4	no	—
117	#543	—	10	4	no	—
118	#533	—	5	8	no	—
119	#533	—	10	8	no	—
120	C₂×#117	—	5	4	no	—
121	C₂×#117	C₂×D₅	1	1	yes	characteristic
122	#501	—	5	32	no	—
123	#500	—	5	8	no	—
124	#499	—	5	8	no	—
125	#499	—	5	8	no	—
126	#498	—	5	8	no	—
127	#428	—	5	16	no	—
128	#427	—	5	4	no	—
129	#427	—	5	8	no	—
130	#427	—	5	4	no	—
131	#427	—	5	4	no	—
132	#427	—	5	4	no	—
133	C₂×#107	—	5	4	no	—
134	C₂×#106	—	5	4	no	—
135	#405	C₂×D₅	1	4	yes	—
136	#405	C₂×D₅	1	4	yes	—
137	C₂×#97	C₂×D₅	1	2	yes	—
138	C₂×C₄×C₈	C₂×D₅	1	2	yes	—
139	C₂³×C₅	—	4	2	no	—
140	C₂³×C₅	C₄⋊Q₈	1	4	yes	—
141	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
142	C₂³×C₅	C₂²×D₄	1	2	yes	—
143	C₂×C₄×C₅	—	4	4	no	—
144	C₂×C₄×C₅	C₄⋊Q₈	1	4	yes	—
145	C₂×C₄×C₅	C₄⋊Q₈	1	8	yes	—
146	C₂×C₄×C₅	C₂²×D₄	1	8	yes	—
147	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
148	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₅	C₂²×Q₈	1	2	yes	—
150	C₂×C₄×C₅	C₂²×Q₈	1	4	yes	—
151	C₂×C₄×C₅	C₂²×Q₈	1	2	yes	—
152	C₂×C₄×C₅	C₂²×Q₈	1	4	yes	—
153	C₂×C₄×C₅	C₂⁵	1	1	yes	characteristic
154	C₂×Dic₅	—	2	16	no	—
155	C₂×Dic₅	—	8	8	no	—
156	Dic₁₀	—	8	16	no	—
157	C₅×C₈	C₂²×Q₈	1	2	yes	—
158	C₅×C₈	C₂²×Q₈	1	4	yes	—
159	C₅×C₈	C₂²×Q₈	1	2	yes	—
160	C₅×C₈	C₂²×Q₈	1	8	yes	—
161	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
162	C₂×M₁₆	Dic₁₀	1	8	yes	complemented
163	C₂×M₁₆	C₂²×D₅	1	4	yes	—
164	C₂²×C₈	Dic₁₀	1	8	yes	complemented
165	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
166	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
167	C₂²×C₈	C₂²×D₅	1	2	yes	—
168	C₂²×C₈	C₂²×D₅	1	2	yes	—
169	C₂²×C₈	C₂²×D₅	1	2	yes	—
170	C₄⋊Q₈	—	5	16	no	—
171	C₄⋊Q₈	—	10	16	no	—
172	#125	—	5	16	no	—
173	#125	—	10	16	no	—
174	C₂²⋊Q₈	—	10	16	no	—
175	C₂²⋊Q₈	—	10	32	no	—
176	C₄×Q₈	—	10	16	no	—
177	#117	—	5	8	no	—
178	#117	—	5	8	no	—
179	#117	—	10	4	no	—
180	#117	—	10	4	no	—
181	#117	D₂₀	1	4	yes	—
182	#117	C₂²×D₅	1	4	yes	—
183	C₂×C₄⋊C₄	—	5	8	no	—
184	C₂×C₄⋊C₄	D₂₀	1	2	yes	—
185	C₂×C₂²⋊C₄	—	10	4	no	—
186	C₂×C₂²⋊C₄	D₂₀	1	2	yes	—
187	C₂×C₄²	—	10	4	no	—
188	C₂×C₄²	C₂²×D₅	1	2	yes	—
189	#107	—	5	8	no	—
190	#107	—	5	8	no	—
191	#107	—	5	16	no	—
192	#106	—	5	8	no	—
193	#106	—	5	8	no	—
194	#106	—	5	16	no	—
195	#97	C₂²×D₅	1	8	yes	—
196	C₄×C₈	C₂²×D₅	1	4	yes	—
197	C₄×C₈	C₂²×D₅	1	4	yes	—
198	C₂²×C₅	—	2	4	no	—
199	C₂²×C₅	—	2	2	no	—
200	C₂²×C₅	—	4	4	no	—
201	C₂²×C₅	C₂×C₄⋊Q₈	1	2	yes	—
202	C₂²×C₅	C₂×C₄⋊Q₈	1	4	yes	—
203	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
204	C₄×C₅	—	2	8	no	—
205	C₄×C₅	C₂×C₄⋊Q₈	1	4	yes	—
206	C₄×C₅	C₂×C₄⋊Q₈	1	2	yes	—
207	C₄×C₅	C₂³×Q₈	1	1	yes	characteristic
208	C₄×C₅	C₂³×Q₈	1	1	yes	characteristic
209	Dic₅	—	4	16	no	—
210	C₂⁴	C₂×D₂₀	1	1	yes	characteristic
211	C₂×Q₈	—	10	16	no	—
212	C₂²×C₄	—	10	8	no	—
213	C₂²×C₄	C₂×Dic₁₀	1	4	yes	—
214	C₂²×C₄	C₂×Dic₁₀	1	4	yes	—
215	C₂²×C₄	C₂×D₂₀	1	4	yes	—
216	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₂₀	1	2	yes	—
218	C₂²×C₄	C₂³×D₅	1	2	yes	—
219	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
220	M₁₆	C₂×Dic₁₀	1	16	yes	—
221	C₂×C₈	C₂×Dic₁₀	1	4	yes	—
222	C₂×C₈	C₂×Dic₁₀	1	4	yes	—
223	C₂×C₈	C₂×Dic₁₀	1	8	yes	—
224	C₂×C₈	C₂×Dic₁₀	1	8	yes	—
225	C₂×C₈	C₂×Dic₁₀	1	8	yes	—
226	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
227	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
228	C₂×C₈	C₂³×D₅	1	4	yes	—
229	C₂×C₈	C₂³×D₅	1	2	yes	—
230	C₄⋊C₄	—	5	16	no	—
231	C₄⋊C₄	—	10	8	no	—
232	C₄⋊C₄	—	10	32	no	—
233	C₄⋊C₄	—	10	16	no	—
234	C₄⋊C₄	C₂×D₂₀	1	8	yes	—
235	C₂²⋊C₄	—	10	8	no	—
236	C₂²⋊C₄	—	10	8	no	—
237	C₂²⋊C₄	C₂×D₂₀	1	8	yes	—
238	C₄²	—	10	8	no	—
239	C₄²	—	10	8	no	—
240	C₄²	C₂×D₂₀	1	4	yes	—
241	C₄²	C₂³×D₅	1	4	yes	—
242	C₂×C₅	—	2	4	no	—
243	C₂×C₅	—	2	2	no	—
244	C₂×C₅	C₂²×C₄⋊Q₈	1	1	yes	characteristic
245	C₂×C₅	#3584	1	2	yes	—
246	C₂³	—	4	2	no	—
247	C₂³	#4015	1	4	yes	—
248	C₂³	C₂²×D₂₀	1	1	yes	characteristic
249	C₂³	C₂²×D₂₀	1	2	yes	—
250	Q₈	—	40	16	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	10	16	no	—
253	C₂×C₄	—	40	8	no	—
254	C₂×C₄	#4015	1	4	yes	—
255	C₂×C₄	#4015	1	8	yes	—
256	C₂×C₄	C₂²×Dic₁₀	1	4	yes	—
257	C₂×C₄	C₂²×Dic₁₀	1	2	yes	—
258	C₂×C₄	C₂²×Dic₁₀	1	2	yes	—
259	C₂×C₄	C₂²×Dic₁₀	1	4	yes	—
260	C₂×C₄	C₂²×D₂₀	1	8	yes	—
261	C₂×C₄	C₂²×D₂₀	1	2	yes	—
262	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
263	C₂×C₄	C₂⁴×D₅	1	1	yes	characteristic
264	C₈	C₂²×Dic₁₀	1	2	yes	—
265	C₈	C₂²×Dic₁₀	1	2	yes	—
266	C₈	C₂²×Dic₁₀	1	4	yes	—
267	C₈	C₂²×Dic₁₀	1	8	yes	—
268	C₅	#60660	1	1	yes	characteristic, complemented, Sylow
269	C₂²	—	2	2	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	4	4	no	—
272	C₂²	C₂×#4015	1	2	yes	—
273	C₂²	C₂×#4015	1	4	yes	—
274	C₂²	C₂³×D₂₀	1	1	yes	characteristic
275	C₄	—	2	8	no	—
276	C₄	—	20	16	no	—
277	C₄	C₂×#4015	1	4	yes	—
278	C₄	C₂×#4015	1	2	yes	—
279	C₄	C₂³×Dic₁₀	1	1	yes	characteristic
280	C₄	C₂³×Dic₁₀	1	1	yes	characteristic
281	C₂	—	2	2	no	—
282	C₂	—	2	4	no	—
283	C₂	C₂²×#4015	1	1	yes	characteristic
284	C₂	#10619064	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₀
commutator, cosocle	C₂×C₄×C₅	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₅
Fitting	C₂×C₅×#405	C₂
socle	C₂²×C₅	C₂³×D₄
prime radical	C₂⁴×C₅	C₂×D₄
hypercenter	C₂×#405	D₅
hypocenter	C₅	#60660
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ C₂×#405

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
5. #10610277	C₂	semidirect
6. #10610276	C₂	semidirect
7. #10610210	C₂	semidirect
8. #10610173	C₂	semidirect
30. #64991	C₂²	semidirect
31. #64972	C₂²	semidirect
32. #64972	C₂²	semidirect
33. #64971	C₂²	semidirect
34. #64969	C₂²	semidirect
37. C₂×C₅×M₁₆	Q₈	semidirect
39. C₂²×C₅×C₈	Q₈	semidirect
162. C₂×M₁₆	Dic₁₀	semidirect
164. C₂²×C₈	Dic₁₀	semidirect
268. C₅	#60660	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#10617626	2
C₂×C₅×#405	2
#10615483	2
C₂×#65395	2
#10610277	2
#10610276	2
#10610210	2
#10610173	2
#60660	5